Concentration notation

Brackets, concentration notation, 151 Brass, 311 Bromate ion, 360 Bromine... 

Using a molar concentration notation instead of molahty (c, = nj V/, cf appendix 1), equation 8.264 is translated into... 

Quantum chemists have devised efficient short-hand notation schemes to denote the basis set aseti in an ab initio calculation, although this does mean that a proliferation of abbrevia-liijii.s and acronyms are introduced. However, the codes are usually quite simple to under-sland. We shall concentrate on the notation used by Pople and co-workers in their Gaussian aerie-, of programs (see also the appendix to this chapter). 

Molar concentrations are used so frequently that a symbolic notation is often used to simplify its expression in equations and writing. The use of square brackets around a species indicates that we are referring to that species molar concentration. Thus, [Na ] is read as the molar concentration of sodium ions. ... 

Potentiometric electrochemical cells are constructed such that one of the half-cells provides a known reference potential, and the potential of the other half-cell indicates the analyte s concentration. By convention, the reference electrode is taken to be the anode thus, the shorthand notation for a potentiometric electrochemical cell is... 

A typical Ag/AgCl electrode is shown in figure 11.9 and consists of a silver wire, the end of which is coated with a thin film of AgCl. The wire is immersed in a solution that contains the desired concentration of KCl and that is saturated with AgCl. A porous plug serves as the salt bridge. The shorthand notation for the cell is... 

The kinetic chain length u may also be viewed as merely a cluster of kinetic constants and concentrations which was introduced into Eq. (6.54) to simplify the notation. As an alternative, suppose we define for the purposes of this chapter a fraction p such that... 

In writing Eqs. (7.1)-(7.4) we make the customary assumption that the kinetic constants are independent of the size of the radical and we indicate the concentration of all radicals, whatever their chain length, ending with the Mj repeat unit by the notation [Mj ], This formalism therefore assumes that only the nature of the radical chain end influences the rate constant for propagation. We refer to this as the terminal control mechanism. If we wished to consider the effect of the next-to-last repeat unit in the radical, each of these reactions and the associated rate laws would be replaced by two alternatives. Thus reaction (7. A) becomes... 

Traditional chemical kinetics uses notation that is most satisfactory in two cases all components are gases with or without an inert buffer gas, or all components are solutes in a Hquid solvent. In these cases, molar concentrations represented by brackets, are defined, which are either constant throughout the system or at least locally meaningful. The reaction quotient Z is defined as... 

Notation. A, B, R, S are participants in the reaction the letters also are used to represent concentrations. 

The TLVs for airborne contaminants are based on the premise that although all chemical substances are toxic at some concentration for some period of time, a concentration exists for all substances from which no toxicity may be expected no matter how often the exposure is repeated. A similar premise holds for substances producing irritation, discomfort and nuisance. In using these limits, items such as excursion factors, ceiling values, "skin" notations, mixtures of substances, and inert material should be considered. These factors are discussed below. 

The notation g- o/7 is the unit tox gram-ion, which represents Avogadro s number (6.02 X 10 ) of ions. The pH is a direct measine of the hydrogen ion concentration and is defined by... 

Hereafter, we shall regularly use the square brackets notation, [ ], to indicate concentration. Thus, we read [Fe+I] as ferric ion concentration. ... 

The notation K is used to emphasize that the scale for K% is based on activities, not simply concentrations. In the gas phase one has available partition functions that can be used to calculate Kt. In solution, however, partition functions are not available. The... 

These representations offer the advantage that one need not argue which of the reagents carries OH or Cl into the transition state. Since that is usually not known, this notation sidesteps the issue. From the Brpnsted-Debye-Huckel equation, we recognize that the concentration of each transition state (and therefore the reaction rate) will vary with ionic strength in proportion to the values of K for the given equation. For the first term we have... 

Only the notation is different from the initial condition used for batch reactors. The subscripts in and out are used for flow reactors. The outlet concentration is found by setting z = L. 

I continue with a few eccentricities in notation, using a, b, c,... to denote molar concentrations of components A, B, C, I have tried to avoid acro-... 

Figure 23 shows sections of with J fixed, C2 = 0 and various values of L/N, where V = 27 is the total polyad action. There is an obvious conical point, X, at K, L) = (7,0). In addition, the concentric contours degenerate to points Y and Z at K, L) = (—7, V). It is important for what follows to explore the character of the point X. As a preliminary, the Hamiltonian may be reduced from the above seven-parameter form, to one involving two essential parameters, both of which vary with the total action. The appropriate definitions [14, 28, 29], modified to conform with the present (7, K) notation, are... 

Concentrations are used so frequently in chemistry that a shorthand notation for concentration is almost essential. Chemists represent the molar concentration of a species by enclosing its formula in brackets ... 

Chemists use logarithmic notation to express this huge range of concentrations. As one example. 

Before comparing these predictions regarding the critical point with experimental results, we may profitably examine the binodial curve of the two-component phase diagram required by theory. The following useful approximate relationship between the composition V2 of the more dilute phase and the ratio y = V2/v2 of the compositions of the two phases may be derived (see Appendix A) by substituting Eq. (XII-26) on either side of the first of the equilibrium conditions (1), using the notation V2 for the volume fraction in the more dilute phase and V2 for that in the more concentrated phase, and similarly substituting Eq. (XII-32) for fX2 and y,2 in the second of these conditions ... 

Consider a few examples where the equilibrium concentrations of components of one of the phases [e.g., (y)] are calculated when their concentrations in the other phase (a) are given. Assumptions will be made that simplify the notation we assume that the activities of the ions appearing in the thermodynamic equations are equal to their concentrations, and we consider systems where the ions have like valencies z. 

Thus, all electrochemical reactions can be characterized by the form of kinetic relation and by the set of coefficients k, a (and, if necessary p), and the values of the concentrations Cj. Particular values of the coefficients always hold for specific reactions hence, the corresponding indices should be appended to the coefficients. In the following, when considering the relatively simple redox reaction (6.2), we use the notations a, and for the coefficients and concentration of the anodic reaction (from left to right) and k, p, and for those of the cathodic reaction occurring in the opposite direction. 

The kinetic equations describing the joint effects of activation and concentration polarization are very complex and we shall consider only the the case of a simple first-order reaction of the type (6.2) proceeding in the presence in the solntion of an excess of a foreign electrolyte. To simplify the appearance of these equations (which even in this case are very cnmbersome), in this section we use a more compact notation that contains two new kinetic parameters ... 

At low values of the bulk concentration Bcy surface coverage is proportional to this concentration, but at high values it tends toward a limit of unity. This equation was derived by Irving Langmuir in 1918 with four basic assumptions (1) the adsorption is reversible (2) the number of adsorption sites is limited, and the value of adsorption cannot exceed A° (3) the surface is homogeneous aU adsorption sites have the same heat of adsorption and hence, the same coefficient B and (4) no interaction forces exist between the adsorbed particles. The rate of adsorption is proportional to the bulk concentration and to the fraction 1-9 of vacant sites on the surface = kjil - 9), while the rate of desorption is proportional to the fraction of sites occupied Vj = kjd. In the steady state these two rates are equal. With the notation kjk = B, we obtain Eq. (10.14). 

From now on, we adopt a notation that reflects the chemical nature of the data, rather than the statistical nature. Let us assume one attempts to analyze a solution containing p components using UV-VIS transmission spectroscopy. There are n calibration samples ( standards ), hence n spectra. The spectra are recorded at q wavelengths ( sensors ), digitized and collected in an nx.q matrix S. The information on the known concentrations of the chemical constituents in the calibration set is stored in an nxp matrix C. Each column of C contains the concentrations of one of the p analytes, each row the concentrations of the analytes for a particular calibration standard. 

In many cases, one may measure spectra of solutions of the pure components directly, and the above estimation procedure is not needed. For the further development of the theory of multicomponent analysis we will therefore abandon the hat-notation in K. Given the pure spectra, i.e. given K (pxq), one may try and estimate the vector of concentrations (pxl) of a new sample from its measured... 

At this point we introduce the formal notation, which is commonly used in literature, and which is further used throughout this chapter. In the new notation we replace the parameter vector b in the calibration example by a vector x, which is called the state vector. In the multicomponent kinetic system the state vector x contains the concentrations of the compounds in the reaction mixture at a given time. Thus x is the vector which is estimated by the filter. The response of the measurement device, e.g., the absorbance at a given wavelength, is denoted by z. The absorbtivities at a given wavelength which relate the measured absorbance to the concentrations of the compounds in the mixture, or the design matrix in the calibration experiment (x in eq. (41.3)) are denoted by h. ... 

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